Hindawi Publishing Corporation Boundary Value Problems Volume 2011, Article ID 684542, 15 pages doi:10.1155/2011/684542 Research Article Minimal Nonnegative Solution of Nonlinear Impulsive Differential Equations on Infinite Interval Xuemei Zhang,1 Xiaozhong Yang,1 and Meiqiang Feng2 Department of Mathematics and Physics, North China Electric Power University, Beijing 102206, China School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, China Correspondence should be addressed to Xuemei Zhang, zxm74@sina.com Received 20 May 2010; Accepted 19 July 2010 Academic Editor: Gennaro Infante Copyright q 2011 Xuemei Zhang et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited The cone theory and monotone iterative technique are used to investigate the minimal nonnegative solution of nonlocal boundary value problems for second-order nonlinear impulsive differential equations on an infinite interval with an infinite number of impulsive times All the existing results obtained in previous papers on nonlocal boundary value problems are under the case of the boundary conditions with no impulsive effects or the boundary conditions with impulsive effects on a finite interval with a finite number of impulsive times, so our work is new Meanwhile, an example is worked out to demonstrate the main results Introduction The theory of impulsive differential equations describes processes which experience a sudden change of their state at certain moments Processes with such a character arise naturally and often, especially in phenomena studied in physics, chemical technology, population dynamics, biotechnology, and economics The theory of impulsive differential equations has become an important area of investigation in the recent years and is much richer than the corresponding theory of differential equations For an introduction of the basic theory of impulsive differential equations in Rn ; see Lakshmikantham et al , Bainov and Simeonov , and Samo˘lenko and Perestyuk and the references therein ı Usually, we only consider the differential equation, integrodifferential equation, functional differential equations, or dynamic equations on time scales on a finite interval with a finite number of impulsive times To identify a few, we refer the reader to 4–13 and references therein In particular, we would like to mention some results of Guo and Liu Boundary Value Problems and Guo In , by using fixed-point index theory for cone mappings, Guo and Liu investigated the existence of multiple positive solutions of a boundary value problem for the following second-order impulsive differential equation: −x t t ∈ J, t / tk , k f t, x t Δx|t I k x tk , tk ax − bx 1, 2, , m, dx 1.1 1, 2, , m, cx θ, k θ, where f ∈ C J × P, P , J 0, , P is a cone in the real Banach space E, θ denotes the zero element of E, f t, θ θ for t ∈ J, Ik θ θ, k 1, 2, , m, < t1 < t2 < · · · < tk < · · · < tm < 1, a ≥ 0, b ≥ 0, c ≥ 0, d ≥ and δ ac ad bc > In , by using fixed-point theory, Guo established the existence of solutions of a boundary value problem for the following second-order impulsive differential equation in a Banach space E : −x t t ∈ J, t / tk , k f t, x, x t Δx|t Δx tk t tk N k x t k , x tk , I k x tk , ax − bx x0 , k 1, 2, , m, 1, 2, , m, k cx 1.2 1, 2, , m, dx ∗ x0 , ∗ where f ∈ C J × E × E, E , J 0, , Ik ∈ C E, E , Nk ∈ C E × E, E , x0 , x0 ∈ E, < t1 < t2 < · · · < tk < · · · < tm < 1, and p ac ad bc / On the other hand, the readers can also find some recent developments and applications of the case that impulse effects on a finite interval with a finite number of impulsive times to a variety of problems from Nieto and Rodr´guez-Lopez 14–16 , ı ´ Jankowski 17–19 , Lin and Jiang 20 , Ma and Sun 21 , He and Yu 22 , Feng and Xie 23 , Yan 24 , Benchohra et al 25 , and Benchohra et al 26 Recently, in 27 , Li and Nieto obtained some new results of the case that impulse effects on an infinite interval with a finite number of impulsive times By using a fixed-point theorem due to Avery and Peterson 28 , Li and Nieto considered the existence of multiple positive solutions of the following impulsive boundary value problem on an infinite interval: u t q t f t, u Δu tk 0, ∀0 < t < ∞, t / tk , k I k u tk , m−2 αi u ξi , u k 1, 2, , p 1, 2, , p, u ∞ 1.3 0, i where f ∈ C 0, ∞ × 0, ∞ , 0, ∞ , Ik ∈ C 0, ∞ , 0, ∞ , u ∞ limt → ξ1 < ξ2 < · · · < ξm−2 < ∞, < t1 < t2 < · · · < < ∞, and q ∈ C 0, ∞ , 0, ∞ ∞u t , 0< Boundary Value Problems At the same time, we also notice that there has been increasing interest in studying nonlinear differential equation and impulsive integrodifferential equation on an infinite interval with an infinite number of impulsive times; to identify a few, we refer the reader to Guo and Liu 29 , Guo 30–32 , and Li and Shen 33 It is here worth mentioning the works by Guo 31 In 31 , Guo investigated the minimal nonnegative solution of the following initial value problem for a second order nonlinear impulsive integrodifferential equation of Volterra type on an infinite interval with an infinite number of impulsive times in a Banach space E: Δx|t Δx ∀t ≥ 0, t / tk , f t, x, T x , x tk I k x tk , N k x tk t tk x0 , x k x 1, 2, , 1.4 ∗ x0 , ∗ where f ∈ C J × P × P, E , Ik ,Nk ∈ C P, P , J 0, ∞ , x0 ,x0 ∈ P, < t1 < · · · < tk < · · · < · · · , tk → ∞, as k → ∞, P is a cone of E However, the corresponding theory for nonlocal boundary value problems for impulsive differential equations on an infinite interval with an infinite number of impulsive times is not investigated till now Now, in this paper, we will use the cone theory and monotone iterative technique to investigate the existence of minimal nonnegative solution for a class of second-order nonlinear impulsive differential equations on an infinite interval with an infinite number of impulsive times Consider the following boundary value problem for second-order nonlinear impulsive differential equation: −x t Δx|t Δx t ∈ J, t / tk , f t, x t , x t tk I k x tk , k 1, 2, , t tk I k x tk , k 1, 2, , ∞ x g t x t dt, x ∞ 1.5 0, 0, ∞ , < t1 < t2 < · · · < tk < where J 0, ∞ , f ∈ C J × R × R , R , R ∞ · · · , tk → ∞, Ik ∈ C R , R , I k ∈ C R , R , g t ∈ C R , R , with g t dt < x ∞ limt → ∞ x t Δx|t tk denotes the jump of x t at t tk , that is, Δx|t tk x tk − x t − , k where x tk and x t− represent the right-hand limit and left-hand limit of x t at t k respectively Δx |t tk has a similar meaning for x t 1.6 tk , Boundary Value Problems Let P C J, R x : x is a map from J into R such that x t is continuous at t / tk , tk and x tk exist for k left continuous at t P C1 J, R x ∈ P C J, R : x t exists and is continuous at t / tk , left continuous at t x 1, 2, , tk and x tk exist for k Let E {x ∈ P C1 J, R : supt∈J |x t |/ max{ x , x ∞ }, where x sup t∈J |x t | , t 1, 2, < ∞, supt∈J |x t | < ∞} with the norm t x 1.7 ∞ sup x t t∈J 1.8 Define a cone P ⊂ E by P x ∈ E : x t ≥ 0, x t ≥ 1.9 Let J J \ {t1 , t2 , , tk , , }, J0 0, t1 , and Ji ti , ti i 1, 2, 3, x ∈ E ∩ C2 J , R is called a nonnegative solution of 1.5 , if x t ≥ 0, x t ≥ and x t satisfies 1.5 0, then boundary value problem 1.5 reduces to the If Ik 0, I k 0, k 1, 2, , g t following two point boundary value problem: −x t x f t, x t , x t 0, x ∞ t ∈ J, 0, 1.10 which has been intensively studied; see Ma 34 , Agarwal and O’Regan 35 , Constantin 36 , Liu 37, 38 , and Yan and Liu 39 for some references along this line The organization of this paper is as follows In Section 2, we provide some necessary background In Section 3, the main result of problem 1.5 will be stated and proved In Section 4, we give an example to illustrate how the main results can be used in practice Preliminaries To establish the existence of minimal nonnegative solution in E of problem 1.5 , let us list the following assumptions, which will stand throughout this paper Boundary Value Problems H1 Suppose that f ∈ C J × R × R , R , Ik ∈ C R , R , I k ∈ C R , R , and there exist p,q,r ∈ C J, R and nonnegative constants ck , dk , ek , fk such that f t, u, v ≤ p t u q t v ∀t ∈ J, and ∀u, v ∈ R , r t , Ik u ≤ c k u dk , ∀u ∈ R k 1, 2, , I k u ≤ ek u fk , ∀u ∈ R k 1, 2, , ∞ p∗ ∞ q∗ dt < ∞, p t t r∗ ∞ ∞ c∗ r t dt < ∞, tk d∗ ∞ 2.1 ck < ∞, k ∞ e∗ dk < ∞, k v2 k q t dt < ∞, tk f∗ ek < ∞, k ∞ fk < ∞ k H2 f t, u1 , v1 ≤ f t, u2 , v2 , Ik u1 ≤ Ik u2 , I k u1 ≤ I k u2 , for t ∈ J, u1 ≤ u2 , v1 ≤ 1, 2, Lemma 2.1 Suppose that H1 holds Then for all x ∈ P , ∞ are convergent k I k x tk ∞ f t, x t , x t dt, ∞ k Ik x tk , and Proof By H1 , we have f t, x t , x t ≤p t t x t t q tx t Ik x tk ≤ c k tk x tk tk dk , I k x tk ≤ e k tk x tk tk r t , fk 2.2 Thus, ∞ f s, x s , x s ds ≤ p∗ ||x||1 ∞ q∗ x ∞ I k x tk ≤ c∗ x d∗ < ∞, I k x tk ≤ e∗ x f ∗ < ∞ k ∞ k The proof is complete r ∗ < ∞, 2.3 Boundary Value Problems ∞ Lemma 2.2 Suppose that H1 holds If ≤ g t dt < 1, then x ∈ E ∩ C2 J , R is a solution of problem 1.5 if and only if x ∈ E is a solution of the following impulsive integral equation: ∞ ∞ G t, s f s, x s , x s ds x t ∞ G t, tk I k x tk Gs t, tk Ik x tk k 1− g t dt ∞ ∞ k ∞ g t ∞ G t, s f s, x s , x s ds 0 G t, tk I k x tk 2.4 k ∞ dt, Gs t, tk Ik x tk ∀t ∈ J, k where ⎧ ⎨t, G t, s ≤ t ≤ s < ∞, ⎩s, ≤ s ≤ t < ∞, ⎧ ⎨0, ≤ t ≤ s < ∞, Gs t, s ⎩1, 2.5 ≤ s ≤ t < ∞ Proof First, suppose that x ∈ E ∩ C2 J , R is a solution of problem 1.5 It is easy to see by integration of 1.5 that −x t t f s, x s , x s ds x 0 I k x tk 2.6 tk